# Introduction to Linear Optimization

# Introduction to Linear Optimization

by Dimitris Bertsimas and John Tsitsiklis

The book is a modern and unified introduction to linear optimization (linear programming, network flows and integer programming) at the PhD level. It covers, in addition to the classical material, all the recent developments in the field in the last ten years including the development of interior points, large scale optimization models and algorithms and complexity of linear optimization. It emphasizes the underlying geometry, intuition and applications of large scale systems.

### Educational Philosophy

Our general philosophy is that insight matters most. For the subject matter of this book, this necessarily requires a geometric view. On the other hand, problems are solved by algorithms, and these can only be described algebraically. Hence, our focus is on the beautiful interplay between algebra and geometry. We build understanding using figures and geometric arguments, and then translate ideas into algebraic formulas and algorithms. Given enough time, we expect that the reader will develop the ability to pass from one domain to the other without much effort.

Another of our objectives is to be comprehensive, but economical. We have made an effort to cover and highlight all of the principal ideas in this field. However, we have not tried to be encyclopedic, or to discuss every possible detail relevant to a particular algorithm. Our premise is that once mature understanding of the basic principles is in place, further details can be acquired by the reader with little additional effort.

Our last objective is to bring the reader up to date with respect to the state of the art. This is especially true in our treatment of interior point methods, large scale optimization, and the presentation of case studies that stretch the limits of currently available algorithms and computers.

### Distinguishing Characteristics

The success of any optimization methodology hinges on its ability to deal with large and important problems. In that sense, the last chapter, on the art of linear optimization, is a critical part of this book.

It will, we hope, convince the reader that progress on challenging problems requires both problem specific insight, as well as a deeper understanding of the underlying theory.

In any book dealing with linear programming, there are some important choices to be made regarding the treatment of the simplex method. Traditionally, the simplex method is developed in terms of the full simplex tableau, which tends to become the central topic. We have found that the full simplex tableau is a useful device for working out numerical examples. But other than that, we have tried not to overemphasize its importance.

Let us also mention another departure from many other textbooks. Introductory treatments often focus on standard form problems, which is sufficient for the purposes of the simplex method. On the other hand, this approach often leaves the reader wondering whether certain properties are generally true, and can hinder the deeper understanding of the subject. We depart from this tradition: we consider the general form of linear programming problems and define key concepts (e.g., extreme points) within this context. (Of course, when it comes to algorithms, we often have to specialize to the standard form.) In the same spirit, we separate the structural understanding of linear programming from the particulars of the simplex method. For example, we include a derivation of duality theory that does not rely on the simplex method.

Finally, this book contains a treatment of several important topics that are not commonly covered. These include a discussion of the column geometry and of the insights it provides into the efficiency of the simplex method, the connection between duality and the pricing of financial assets, a unified view of delayed column generation and cutting plane methods, stochastic programming and Benders decomposition, the auction algorithm for the assignment problem, certain theoretical implications of the ellipsoid algorithm, a thorough treatment of interior point methods, and a whole chapter on the practice of linear optimization. There are also several noteworthy topics that are covered in the exercises, such as Leontief systems, strict complementarity, options pricing, von Neumann's algorithm, sub-modular function minimization, and bounds for a number of integer programming problems.

This book includes themes of modeling, complexity, and algorithmic approaches. There are a fair number of exercises that are given at the end of each chapter. Most of them are intended to deepen the understanding of the subject, or to explore extensions of the theory in the text, as opposed to routine drills. However, several numerical exercises are also included. Starred exercises are supposed to be fairly hard.

### For Instructors

A solution manual is available to qualified instructors upon request. Please fill out the form in the "For Instructors" section under Resources.